Gromov-Witten invariants of log Calabi-Yau 3-folds are holomorphic lagrangian correspondences

TL;DR

This paper develops a holomorphic symplectic category with objects as holomorphic symplectic manifolds and morphisms as lagrangian correspondences, linking Gromov-Witten invariants of log Calabi-Yau 3-folds to these correspondences.

Contribution

It introduces a holomorphic symplectic category framework and encodes Gromov-Witten invariants of log Calabi-Yau 3-folds as lagrangian correspondences, proposing a new geometric perspective.

Findings

Gromov-Witten invariants are encoded as holomorphic lagrangian correspondences.

Extension of the symplectic category to log schemes.

Conjectural relation between Gromov-Witten and Donaldson-Thomas invariants.

Abstract

We introduce a holomorphic version of Weinstein's symplectic category, in which objects are holomorphic symplectic manifolds, and morphisms are holomorphic lagrangian correspondences. We then extend this category to log schemes, and prove that Gromov-Witten invariants of log Calabi-Yau 3-folds are naturally encoded as holomorphic lagrangian correspondences. Gromov-Witten invariants and Donaldson-Thomas invariants are then conjecturally related by a natural unitary lagrangian correspondence.